Summary

I compare the hierarchical risk parity (HRP) portfolio to six other portfolio strategies: inverse volatility (IV), equal weight (EW), global minimum variance (GMV), maximum diversification (MD) equal risk contribution (ERC) and maximum sharpe ratio (MSR). I report out-of sample performance, portfolio composition and correlation in two asset sets: S&P500 stocks from Dec 1991 thru Nov 2016 and in global futures from Jan 2002 thru Apr 2016. I simulate both the original version of the HRP portfolio using variance risk measure (HRP-VAR) as well as a similarly defined portfolio using the volatility risk measure (HRP-VOL).

HRP-VOL turns out to be very closely related the ERC portfolio. Performances are highly correlated, and both portfolios allocate funds in similar proportions to assets and reach a similar performance. Both are essentially mechanisms to allocate a share of volatility risk to every asset, leading to the same benefits (balanced allocation, smaller dependence on estimation error) and drawbacks (sub-optimal diversification). Furthermore, the benefits of the HRP portfolio claimed by the author - reduced sensitivity to parameter estimation leading to increased robustness in large asset sets - apply to both strategies and do not discriminate between them.

HRP-VAR, the original hierarchical risk parity portfolio based on variance risk, outperforms its declared benchmark, the GMV portfolio, in global futures. This is of limited importance, as the GMV portfolio performs poorly in this data set in the first place and since it concentrates the allocation in a small number of assets with lowest variance. The outperformance is caused by a a more balanced allocation of the HRP-VAR portfolio. I imagine that an ERC portfolio based on the variance risk measure would lead to a similar outcome.

While I sympathize with the algorithmic approach of the portfolio, I do not see a clear benefit over the already established and much better researched alternatives.

Allocation Strategies

Hierarchical Risk Parity (HRP)

López de Prado (2016) orders assets hierarchically and then recursively cuts asset in halves, allocating an equal share of risk to both halfs.

  1. Hierarchically cluster assets based on distance \(d_{i,j}\) \[ d_{i,j} = \sqrt{\frac{1}{2}(1-\rho_{i,j})}\quad\quad \rho_{i,j}\text{: correlation between assets } i \text{ and } j \]
  2. Order assets in sequence of the dendogramm, see figures 1-4 for illustration
  3. Allocate risk through recursive bi-section:
    • split assets into halves
    • assign equal variance \(\sigma^2 = \mathbf{w'\Sigma w}\) using \(w_i = 1 / \sigma_i^2\) to each half, and
    • repeat above steps for each half of the assets.

He claims that this algorithm reduces sensitivity to paramter estimation, leads to more stable portfolios and is particularely well suited for a large number of portfolio constituents.

López de Prado (2016) uses the variance risk measure \(\sigma^2\) and uses the global minimum variance (GMV) portfolio as a benchmark. I name his definition HRP-VAR and introduce a similar portfolio, HRP-VOL, based on volatility instead of variance. Risk is now measured as \(\sigma = \sqrt{\mathbf{w'\Sigma w}}\) using \(w_i = 1 / \sigma_i\).

Dendrogram

Figures 1-4 show the dendrogram of hierarchical clusters computed with global futures at four moments throughout our study period. Clusters are based on the previous 3 years of returns. Colors assign futures instruments to asset classes equities, commodities and fixed income.

Observations:

  • The hierarchical clustering nicely arranges asset of the same asset class side-by side.
  • Assets and entire clusters can, however, switch positions from one iteration to the next.
  • Splitting assets in two halves assigns assets arbitrarily to two halves, once time grouping equities and half the commodities together, another time grouping half the equities and fixed income together.
2002-01-31
<span style="color:grey"> Figure 1: Hierarchical clustering of Global Futures as of 2002-01-31 based on three years of daily returns.</grey>

Figure 1: Hierarchical clustering of Global Futures as of 2002-01-31 based on three years of daily returns.

2006-10-31
<span style="color:grey"> Figure 2: Hierarchical clustering of Global Futures as of 2006-10-31 based on three years of daily returns.</grey>

Figure 2: Hierarchical clustering of Global Futures as of 2006-10-31 based on three years of daily returns.

2011-07-29
<span style="color:grey"> Figure 3: Hierarchical clustering of Global Futures as of 2011-07-29 based on three years of daily returns.</grey>

Figure 3: Hierarchical clustering of Global Futures as of 2011-07-29 based on three years of daily returns.

2016-04-29
<span style="color:grey"> Figure 4: Hierarchical clustering of Global Futures as of 2016-04-29 based on three years of daily returns.</grey>

Figure 4: Hierarchical clustering of Global Futures as of 2016-04-29 based on three years of daily returns.

Other objectives

  • The maximum decorrelation (MDC) portfolio minimizes portfolio variance under the assumption of identical volatility for all assets.

\[ \min_{w \geq 0} \mathbf{w}'\mathbf{P}\mathbf{w} \]

  • The inverse volatility (IV) portfolio scales assets to equal single-asset volatility. \[ w_i = \frac{1}{\sigma_r} \]

  • The maximum diversified (MD) portfolio proposed by Choueifaty and Coignard (2008) maximizes the diversification ratio defined as the ratio between weighted sum of single asset volatility over portfolio volatility. \[ \max_{\mathbf{w}\geq 0} \text{DR}, \quad \text{DR} = \frac{\textbf{w}'R(\textbf{r})}{R(\textbf{w}'\textbf{r})} = \frac{\mathbf{w}'\mathbf{\sigma}_r}{\sqrt{\mathbf{w}'\mathbf{\Sigma}\mathbf{w}}} \]

  • The equal risk contribution (ERC) portfolio formalized by Maillard, Roncalli, and Teïletche (2010) aligns the contribution of each asset to the total portfolio risk. \[ \text{RC}_j = \text{RC}_i = w_i\frac{\partial R(\mathbf{w}'\mathbf{r})}{\partial w_i} = \frac{1}{2} w_i (\mathbf{\Sigma}\mathbf{w})_i \]

  • The equal weight (EW) portfolio allocates an equal amount to all assets: \[ w_i = \frac{1}{N} \]

  • The Maximum sharpe ratio (MSR) portfolio maximizes the portfolio’s excess return divided by volatility: \[ \max_{w\geq 0} \frac{\mathbf{w}'\;\mathbb{E}\left[\mathbf{r} - r_f\right]}{\sqrt{\mathbf{w}'\Sigma\mathbf{w}}} \]

Scaling

Portfolios are calibrated subject to two alternative investment constraints:

  • Fully invested \[ s.t.\;\;\mathbf{w}'\mathbf{1}=1 \]
  • Constant 10% annualized target volatility \[ s.t.\;\;\sqrt{\mathbf{w}'\mathbf{\Sigma}\mathbf{w}} = \sigma_{\text{target}} \]

Results

Risk and Performance

Figures 5-8 and tables 3-5 report out-of-sample risk and performance of the portfolio strategies in both asset sets. Portfolios are calibrated monthly based on the last three years of underlying market returns. Stock returns are reported in excess of the 6-monht T-bill rate. Futures returns are considered to be excess returns. Execution costs are not taken into account.

Observations:

  • The HRP-VAR portfolio outperforms its benchmark, the GMV portfolio, in the global futures data set. Performance in single stocks is very similar between the two.
  • The HRP-VOL portfolio performs similarly as the ERC portfolio. This is not surprising since both effectively allocate equal risk contribution to assets.

Global Futures - 10% Target Vola

Figure 5: Rolling one-year out-of-sample return and volatility. Execution cost is not taken into account. The plot is interactive, lines can be removed/restored by selecting the legend.

<span style="color:grey"> Table 1: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Execution cost is not taken into account. MSE reports the mean squared difference to the 10% volatility target in above figure. </grey>

Table 1: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Execution cost is not taken into account. MSE reports the mean squared difference to the 10% volatility target in above figure.

Single Stocks - 10% Target Vola

Figure 6: Rolling one-year out-of-sample volatility and return in excess of the 6 month T-bill rate. Execution cost is not taken into account. The plot is interactive, lines can be removed/restored by selecting the legend.

<span style="color:grey"> Table 2: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Annual return is in excess of the 6-month T-bill rate. Execution cost is not taken into account. MSE reports the mean squared difference to the 10% volatility target in above figure.</grey>

Table 2: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Annual return is in excess of the 6-month T-bill rate. Execution cost is not taken into account. MSE reports the mean squared difference to the 10% volatility target in above figure.

Single Stocks - Full Invest

Figure 7: Rolling one-year out-of-sample volatility and return in excess of the 6 month T-bill rate. Execution cost is not taken into account. The plot is interactive, lines can be removed/restored by selecting the legend.

<span style="color:grey"> Table 3: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Annual return is reported in excess of the 6-month T-bill rate. Execution cost is not taken into account.</grey>

Table 3: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Annual return is reported in excess of the 6-month T-bill rate. Execution cost is not taken into account.

Correlation

Figures 8-10 below report the pairwise correlation coefficients between out-of-sample returns of the simulated portfolio strategies over the entire study period.

Observations:

  • Global futures universe:
    • The HRP-VAR portfolio highly correlates with the GMV portfolio, this is not surprising given that both heavily load on fixed income assets with low variance.
    • The HRP-VOL portfolio highly correlated with IV and ERC portfolios, which also equally distribute volatility risk to all assets.
  • Single US Stock universe:
    • Both HRP portfolios highly correlate IV, EW and ERC portfolios. This is an indication that all portfolios roughly allocate the same amount of risk to all assets.

Global Futures - 10% Target Vola

<span style="color:grey"> Figure 8: Pairwise correlation between daily returns of portfolio strategies from Jan 2002 thru Apr 2016 in Global Futures scaled to 10% target volatility.</grey>

Figure 8: Pairwise correlation between daily returns of portfolio strategies from Jan 2002 thru Apr 2016 in Global Futures scaled to 10% target volatility.

Single Stocks - 10% Target Vola

<span style="color:grey"> Figure 9: Pairwise correlation between daily out-of-sample returns of portfolio strategies from Dec 1991 thru Nov 2016 in S&P500 Stocks scaled to 10% target volatility.</grey>

Figure 9: Pairwise correlation between daily out-of-sample returns of portfolio strategies from Dec 1991 thru Nov 2016 in S&P500 Stocks scaled to 10% target volatility.

Single Stocks - Full Invest

<span style="color:grey"> Figure 10: Pairwise correlation between daily out-of-sample returns of portfolio strategies from Dec 1991 thru Nov 2016 in S&P500 Stocks scaled to full investment.</grey>

Figure 10: Pairwise correlation between daily out-of-sample returns of portfolio strategies from Dec 1991 thru Nov 2016 in S&P500 Stocks scaled to full investment.

Sensitivity to parameter estimation

Figure 11 below reports the distribution of sharpe ratios of various portfolios over the entire study period when calibrated with 50 bootstrapped calibration periods. Calibration periods are generated by randomly drawing 36 month long blocks of returns out of the original 36-month calibration period.

Observations:

  • We use the variation of outcomes as a proxy for the sensitivity to parameter estimation. The HRP-VAR portfolio has a slightly lower variation than the GMV portfolio, especially in the single US stock universe. The difference in variation is very strong when using the full investment constraint.
  • Variability for the HRP-VOL portfolio is larger than for IV and ERC portfolios, which also assign a similar volatility budget to assets.

Figure 11: Distribution of out-of-sample sharpe ratios over entire study period when each portfolio strategy is calibrated with 50 bootstrapped calibration windows. Calibration periods are generated by randomly drawing 36 month long blocks of returns out of the original 36-month calibration period.

Portfolio Composition

Average Single Asset Volatility

Exposure to low volatility assets is an important performance driver during the study period, during which low volatility assets earn a premium over high volatility assets due to falling interest rates and possibly other factors. At the same time, portfolios with a large exposure to low volatility assets are more exposed to the risk of rising interest rates in the future.

Figures 12 and 13 report the weighted average of single asset volatility held in each portfolio, where volatility \(\sigma_i\) of asset \(i\) is the standard deviation of the asset’s returns during the portfolio calibration period. \[ \text{avg. asset volatlitiy} = \frac{\mathbf{w}'\mathbf{\sigma_r}}{\mathbf{w}'\mathbf{1}} \]

Observations:

  • HRP-VOL allocates very similarly to lower- and higher volatility instruments as IV and ERC strategies, which also distribute a similar volatility budget to all assets.
  • HRP-VAR has similar allocation to lower- and higher volatility instruments as GMV in global futures. The allocation lies between GMV and IV in single US stocks.

Global Futures

Figure 12: Average single asset volatility weighted by proportion held in portfolio. The plot is interactive, lines can be removed/restored by selecting the legend.

S&P500 Stocks

Figure 13: Average single asset volatility weighted by proportion held in portfolio. The plot is interactive, lines can be removed/restored by selecting the legend.

Allocation to Underlying Assets

Figures 9-31 illustrate the portfolio’s allocations to underlying assets. For this illustration, portfolios are calibrated for a subset of 10 randomly selected instruments to keep the number of assets visually digestible.

The procedure to generate random subset avoids introducing a survivorship bias:

  1. Randomly pick ten assets available at the first rebalancing moment.
  2. Stocks exluded from S&P500 membership are replace at the subsequent rebalancing moment by a randomly picked stock which has entered the universe since the preceding rebalancing moment.

Observations:

  • The HRP-VAR portfolio is a better balanced version of the GMV portfolio with still high but nevertheless less extreme allocation to low variance assets.
  • The HRP-VOL portfolio has a very similar allocation as the ERC portfolio. Clearly, the two are closely related.

Global Futures

Figures 14-21 report the fraction of funds allocated to underlying assets by the different portfolio strategies, figure 22 reports the rolling volatility of portfolio constituents during the calibration period.

HRP-VAR
<span style="color:grey"> Figure 14: Fraction of funds allocated to underlying assets by the hierarchical risk parity with variance (HRP-VAR) portfolio.</grey>

Figure 14: Fraction of funds allocated to underlying assets by the hierarchical risk parity with variance (HRP-VAR) portfolio.

HRP-VOL
<span style="color:grey"> Figure 15: Fraction of funds allocated to underlying assets by the hierarchical risk parity with volatility (HRP-VOL) portfolio.</grey>

Figure 15: Fraction of funds allocated to underlying assets by the hierarchical risk parity with volatility (HRP-VOL) portfolio.

EW
<span style="color:grey"> Figure 16: Fraction of funds allocated to underlying assets by the equal weight (EW) portfolio.</grey>

Figure 16: Fraction of funds allocated to underlying assets by the equal weight (EW) portfolio.

MSR
<span style="color:grey"> Figure 17: Fraction of funds allocated to underlying assets by the maximum sharpe ratio (MSR) portfolio.</grey>

Figure 17: Fraction of funds allocated to underlying assets by the maximum sharpe ratio (MSR) portfolio.

IV
<span style="color:grey"> Figure 18: Fraction of funds allocated to underlying assets by the inverse volatility (IV) portfolio.</grey>

Figure 18: Fraction of funds allocated to underlying assets by the inverse volatility (IV) portfolio.

GMV
<span style="color:grey"> Figure 19: Fraction of funds allocated to underlying assets by the global minimum variance (GMV) portfolio.</grey>

Figure 19: Fraction of funds allocated to underlying assets by the global minimum variance (GMV) portfolio.

MD
<span style="color:grey"> Figure 20: Fraction of funds allocated to underlying assets by the maximum diversified (MD) portfolio.</grey>

Figure 20: Fraction of funds allocated to underlying assets by the maximum diversified (MD) portfolio.

ERC
<span style="color:grey"> Figure 21: Fraction of funds allocated to underlying assets by the equal risk congtribution (ERC) portfolio.</grey>

Figure 21: Fraction of funds allocated to underlying assets by the equal risk congtribution (ERC) portfolio.

Asset volatility

Figure 22: Volatility of portfolio constituents during the rolling calibration period.

S&P500 Stocks

Figures 23-30 report the fraction of funds allocated to underlying assets by the different portfolio strategies, figure 31 reports the rolling volatility of portfolio constituents during the calibration period.

HRP-VAR
<span style="color:grey"> Figure 23: Fraction of funds allocated to underlying assets by the hierarchical risk parity with variance (HRP-VAR) portfolio.</grey>

Figure 23: Fraction of funds allocated to underlying assets by the hierarchical risk parity with variance (HRP-VAR) portfolio.

HRP-VOL
<span style="color:grey"> Figure 24: Fraction of funds allocated to underlying assets by the hierarchical risk parity with volatility (HRP-VOL) portfolio.</grey>

Figure 24: Fraction of funds allocated to underlying assets by the hierarchical risk parity with volatility (HRP-VOL) portfolio.

EW
<span style="color:grey"> Figure 25: Fraction of funds allocated to underlying assets by the equal weight (EW) portfolio.</grey>

Figure 25: Fraction of funds allocated to underlying assets by the equal weight (EW) portfolio.

MSR
## Warning: Removed 40 rows containing missing values (position_stack).
<span style="color:grey"> Figure 26: Fraction of funds allocated to underlying assets by the maximum sharpe ratio (MSR) portfolio.</grey>

Figure 26: Fraction of funds allocated to underlying assets by the maximum sharpe ratio (MSR) portfolio.

IV
<span style="color:grey"> Figure 27: Fraction of funds allocated to underlying assets by the inverse volatility (IV) portfolio.</grey>

Figure 27: Fraction of funds allocated to underlying assets by the inverse volatility (IV) portfolio.

GMV
<span style="color:grey"> Figure 28: Fraction of funds allocated to underlying assets by the global minimum variance (GMV) portfolio.</grey>

Figure 28: Fraction of funds allocated to underlying assets by the global minimum variance (GMV) portfolio.

MD
<span style="color:grey"> Figure 29: Fraction of funds allocated to underlying assets by the maximum diversified (MD) portfolio.</grey>

Figure 29: Fraction of funds allocated to underlying assets by the maximum diversified (MD) portfolio.

ERC
<span style="color:grey"> Figure 30: Fraction of funds allocated to underlying assets by the equal risk congtribution (ERC) portfolio.</grey>

Figure 30: Fraction of funds allocated to underlying assets by the equal risk congtribution (ERC) portfolio.

Asset volatility

Figure 31: Volatility of portfolio constituents during the rolling calibration period.

References

Choueifaty, Yves, and Yves Coignard. 2008. “Toward Maximum Diversification.” The Journal of Portfolio Management 35 (1): 40–51.

López de Prado, Marcos. 2016. “Building Diversified Portfolios That Outperform Out of Sample.” The Journal of Portfolio Management 42 (4): 59–69.

Maillard, Sébastien, Thierry Roncalli, and Jérôme Teïletche. 2010. “The Properties of Equally Weighted Risk Contribution Portfolios.” The Journal of Portfolio Management 36 (4): 60–70.